We remark that thm:spanning-trees is not the first robust version of the results of Komlós, Sárközy, and Szemerédi; Balogh, Csaba, and Samotij [1]} consider the resilience of \(\mathbb {G}(n,p)\)  with respect to containing almost spanning trees. Specifically, for every \(\Delta \in \mathbb {N}\)  and \(\varepsilon ,\eta >0\) , in \(\mathbb {G}(n,p)\)  with \(p = \Omega _{\eta ,\varepsilon ,\Delta }(1/n)\) , w.h.p. any subgraph \(G\)  with at least \((1/2+\eta )\) -fraction of the edges at each vertex contains any tree on \((1-\varepsilon )n\)  vertices with degree bounded by \(\Delta \) . Related work has also considered such universality with respect to containment of almost spanning structures of limited bandwidth in \(\mathbb {G}(n,p)\)  and resilient subgraphs; we refer the reader to [2]} and references therein.
